Topology

Originally posted by PBE6
Wait a minute, is that true? A double-torus or figure-eight has 2 rings, each with one "in" and one "out". But a y-tube has 3 "holes", each with one "in" and two "outs". I don't think you can stretch a figure-eight into a y-tube without tearing the surface or eliminating some holes.

It's a bit hard to describe in words (use your imagination!) but you can get from a Y-tube to an 8 by homotopy, as follows:

- collapse the Y-tube into the 'junction' in the middle, which is basically a pair of triangles, with each corner of one triangle connected by a line to a corner of the other triangle
- pick one of the lines, and pull the line itself and both triangles onto its midpoint. Stretch the other two lines over where the triangle used to be, so that everything is continuous. You now have a pair of loops connected at one point, ie an 8.

Originally posted by Acolyte
It's a bit hard to describe in words (use your imagination!) but you can get from a Y-tube to an 8 by homotopy, as follows:

- collapse the Y-tube into the 'junction' in the middle, which is basically a pair of triangles, with each corner of one triangle connected by a line to a corner of the other triangle
- pick one of the lines, and pull the line it ...[text shortened]... so that everything is continuous. You now have a pair of loops connected at one point, ie an 8.

I don't think that works either. Isn't the idea of homotopy that one shape can become another shape by deformation? Attaching a piece of the shape to another piece of itself isn't a deformation, it's a construction.

Originally posted by Acolyte
Think of an apple where n worms have entered the apple, chewed their way through and come out again, without crossing their own or each others' paths. Say that an solid object is a 'solid torus of genus n' or a 'solid n-holed torus' i ...[text shortened]... 's homotopy equivalent to a figure-of-eight, if that's any help.

Got it. A hole has to have two openings to the outside such that something can go in one end and come out the other. How about going in one nostril and out the other? Do eyes count?

EDIT - Never mind about the nostrils; they are connected to the GI tract.

Originally posted by PBE6
I don't think that works either. Isn't the idea of homotopy that one shape can become another shape by deformation? Attaching a piece of the shape to another piece of itself isn't a deformation, it's a construction.

Think about it the other way. Start with an 8, then stretch the outer perimeter out one way to form the base of the Y tube and each of the circles out the other way to form the arms of the Y tube.

Originally posted by AThousandYoung
EDIT - Never mind about the nostrils; they are connected to the GI tract.

Oh, I didn't think of that. In that case you have three holes going in and one going out, all meeting at a junction in the middle. This is homotopy equivalent to three circles joined at a single point (similar to the Y-tube), but NOT to the worm-ridden apple, for any number of worms.

Originally posted by richjohnson
Think about it the other way. Start with an 8, then stretch the outer perimeter out one way to form the base of the Y tube and each of the circles out the other way to form the arms of the Y tube.

Aha! That makes sense.